Synthetic & Analytic Trigonometry

Trivium

  1. Given the measure of an angle expressed in radians, or as a fraction of a degree, or in degrees/minutes/seconds, express it the other two ways.
  2. Given an origin point and a coordinate axis in a plane, use a protractor and ruler to determine the polar coordinates of any point in that plane.
  3. Given either the rectangular or polar coordinates of a point in a plane write down its coordinates in the other coordinate system.
  4. For a right triangle, given either the lengths of two sides or the length of one side and measure of one angle, compute the lengths of the remaining sides, the measure of the remaining angles, and the area of the triangle.
  5. For a general triangle, given either the lengths of three sides, the lengths of two sides and the measure of an angle, or the lengths of one side and the measure of two angles, determine all possible triangles that have the measurements given, and for those possibilities compute the lengths of the remaining sides, the measure of the remaining angles, and the area of the triangle.
  6. Given enough information to uniquely determine a triangle, use a protractor and ruler to accurately draw that triangle.
  7. Given two angles and their sine and cosine, compute the sine and cosine of the sum and difference of those angles.
  8. Given an angle and its cosine, compute the sine and cosine of half that angle.

Problems

  1. What is the degree measure of an angle that measures thirty-one radians? How do you express this degree measure in degrees/minutes/seconds?
  2. The earth is approximately spherical, with a radius of 3960 miles. Classically, a nautical mile was defined to be the distance along the earth’s surface subtended by a central angle of 1′, i.e. one-sixtieth of a degree. How many miles are in a nautical mile?
  3. Suppose a truck’s tires have a diameter of 28″. If the truck travels one mile, how many times do its tires revolve? I.e. if there is a rock stuck in the tread of one of the tires, how many times will that rock click against the asphalt road over the course of a mile?
  4. three circles mutually tangent to to each other, the segment from the center of the smallest to the centers of the other two highlighted, and the sector of the smallest circle bounded by those segments is highlighted Consider three circles having radii measuring one, two, and three, that are each mutually tangent to each other. What is the area of the sector of the smallest circle that is cut off by the two segments joining its center to the centers of the others?
  5. Mr. Goat is tethered to the concave corner of an “L”-shaped shed. The ground outside the shed is covered with lush delicious ground-cover — grasses and clovers and creeping red thyme — that Mr. Goat would like to eat. an image of how Mr. Goat is tethered to the shed. Idk how to describe it accurately, so I'll work on making this an SVG file instead some day. Considering the length of Mr. Goat’s tether and the dimensions of the shed indicated in the image, what is the total area of ground cover that Mr. Goat can raze? Challenge: Suppose instead of that inner corner, Mr. Goat were tethered to the upper-right corner indicated with a small square. Now what is the total area of ground cover that Mr. Goat can raze?
  6. a compass rose featuring all principal, half-, and quarter-winds. Sticking to the convention that our polar axis points eastward (to the right) and that positive angles correspond to counterclockwise rotation, every direction on a compass rose corresponds to an angle: north is 90°, west is 180°, northeast is 45°, and so on. The directions north (N), south (S), east (E), and west (W) are called the cardinal directions. The directions northeast (NE), southeast (SE), northwest (NW), and southwest (SW) are called the intercardinal directions. Altogether the cardinal and intercardinal directions are referred to as the principal winds. The directions halfway between any two principal winds are called half-winds — e.g. east-northeast (ENE) is the half-wind between east and northeast. The directions halfway between any principal wind and half-wind are called quarter-winds, and are named by saying the nearest principal wind followed by “by” followed by the nearest cardinal direction on its other side — e.g. northeast-by-east (NEbE) is halfway between northeast and east-northeast.
    1. What angle does northeast correspond to? What angle does east-northeast correspond to? What angle does east-by-north correspond to?
    2. Patroclus the Pirate is following a map to buried treasure. Upon landing on the island at the small dilapidated dock indicated on the map, he notices that there’s no mark at the location of the treasure, but instead just this written instruction: Walk 1234 paces* northwest-by-west, and with my treasure you’ll be blessed. Wary of just pacing it out, Patroclus wants to mark the destination on his map first. How many feet north and how many feet west of the dock is the buried treasure?
    3. Roberta has just designed a small walking robot and has taken it out to a soccer field to test it out. This field’s long edge runs east-to-west and short edge runs north-to-south, and the field’s dimensions adhere precisely to FIFA’s recommendation of 105×68 meters. Roberta places the robot at one corner of the field and gives it the instruction to walk to the diagonally opposite corner along the shortest path possible. However, Roberta doesn’t realize that her robot’s navigation system has a design flaw: it can only travel in the directions on a 32-point compass, along the principal, half-, and quarter-winds. The robot still manages to determine the shortest path possible under this constraint and walks to the opposite corner, albeit its path isn’t a straight line. What path did the robot take, and how far was its walk?
  7. What is the slope of the line that passes through the origin and the point on the unit circle with coordinates \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)?\)
  8. What is an equation of the line that passes through the point \((6,7),\) has a positive slope, and forms an acute angle of \(39°\) with the \(x\)-axis?
  9. What is the measure of the acute angle formed at the intersection of the line \(y = \frac{3}{5}x+7\) and the \(x\)-axis? What is the measure of the acute angle formed at the intersection of the lines \(y = \frac{3}{5}x+7\) and \(y = \frac{7}{2}x-3?\) In general, given two lines \(y = m_1x+b_1\) and \(y = m_2x+b_2,\) what is the measure of the angle at which they intersect?
  10. a circle with a segment missing, and arrow pointing to the conical (paper) cup that the circle could be folded into A sector of a circle can be folded up into a cone. Some cheap water cups are constructed this way: start with a disk of waxed paper, make two cuts to remove a small sector of the disk, and adhere the disk along the two cuts to make a “cup”. (Obviously it’s more complicated than this; you need to leave a little paper tab to glue across the cut, but let’s keep this problem simple.) Suppose the initial circular sector has radius \(r\) and has a subtended angle measuring \(\theta.\) What will the height of the corresponding cup be? Challenge: What angle \(\theta\) corresponds to the cup that holds the maximal amount of water?
  11. A rain gutter should be installed at a slight incline, usually called the pitch of the gutter. But “pitch” is just another word for “grade” or “slope”. A common recommendation is that the gutter should drop by ¼″ for every 10′ of roofing. What angle does this recommendation correspond to?
  12. You may have seen road signs like the one here that indicate the road is declined steeply at a specific grade. For example, an 8% grade (decline) indicates that there is an 8% loss in elevation per horizontal distance travelled — e.g. for every 100′ a truck travels horizontally it’ll lose 8′ of elevation. This corresponds directly with the “slope” of the road. Considering the surface of the road as a line, the slope of that line will be -8%.
    1. What angle of declination (depression) does a 6% grade correspond to?
    2. If a road is constructed to have a 6° decline, what is the grade of the road?
    3. If a truck drives along a road with a 6° decline for one mile, how much elevation does it lose?
  13. Angeline is standing on level ground, some distance away from the base of St Mary’s Medical Center, which, at 150′ tall, is the tallest building in Grand Junction. Looking up at the very top of the building, Angeline pulls out her trusty inclinometer and measures that the angle of elevation of her line of sight is 72°. How far away is Angeline from the base of St Mary’s?
  14. Dr Baxter, bored with his daily commute home from work, decides to build a zip line from the roof of St Mary’s Medical Center to the nearby park that is in the direction of his house. St Mary’s is about 150′ tall. He decides he can afford to set up a zip line with a braking system, in which case experts recommend a 6% grade as the steepest direction of decline that is still safe. Sketching out the design, Dr Baxter has to figure out two things: assuming the zip line will be fairly taught, what angle will it make with the level ground in the park? And how long does the line itself need to be?
  15. Celina has climbed a tree! The tree is 33′ tall, growing straight out of level ground. She spots her friend away from the base of the tree and, pulling out her trusty inclinometer, measure the angle of depression towards her friend to be 53°. If her fiend pulls out an inclinometer too and measures the angle of inclination towards Celina, what should the measurement be? How far away from the base of the tree is Celina’s friend?
  16. The Ladder Association advises that, for safety’s sake, when setting up a ladder to be climbed, the base of ladder should make at least a 75° angle with the ground. Demetrius needs to buy a ladder to install rain gutters on his home. He’s about 6′ tall, and the roofline of his house is 14′ off the ground.
    1. If Demetrius is dedicated to always setting up his ladder safely, what’s the shortest possible ladder he should buy?
    2. While out shopping for ladders, Demetrius notices there are painters working on ladders outside the hardware store. He notices that one painter’s ladder is 11′ long and positioned with its feet 3′ from the base of the wall. Should Demetrius go talk to the painter about safety?
    3. Finally looking at ladders in the ladder aisle of the hardware store, Demetrius notices that there are warnings on each ladder to not stand on top, or stand on the top rung. On every ladder for sale, the highest rung that’s safe to stand on is 2′ from the top of the ladder. What’s the shortest possible ladder that Demetrius, still dedicated to safety, should buy?
  17. Edith is flying a kite. She’s let out 123′ of kite string and, using her trusty inclinometer, measures that the kite’s angle of elevation from her position is 36°. Assuming the kite string is taught, what’s the kite’s altitude?
  18. Finneas is hiking to the top of a tall hill that has a constant incline of 25°. After reaching the top of the hill, his altimeter tells him his altitude changed by 6000′ since starting his hike. How long was the hike?
  19. Garamond is standing sad and alone in an empty parking lot. He is 5′10″ tall, and his shadow is 11′ long from his feet to the tip of his shadow-head. What is the angle of inclination of the sun?
  20. Henrietta wants to set up a camera to automatically record an aerial fireworks show. She positions the camera on the level ground about 500′ from where the fireworks will be launched, and Henrietta knows that, on average, these fireworks will launch 200′ in the air. At what angle of inclination should she tilt the camera to ensure she gets a good recording of the show?
  21. An airplane is cruising over the surface of the earth, maintaining a constant altitude. There are two radar antennas on the earth tracking the plane. The moment the plane flies directly over the great circle (geodesic) between them, the two antennas report this information: one reports that the plane is 21 miles away with an angle of elevation of 16.6° and the other antenna reports the plane is 46 miles away with an angle of elevation of 7.5°. Assuming the Earth is flat in the relatively short distance between them, how far apart are the antennas, and what is the plane’s altitude? How do the answers to those questions change if you factor in the curvature of the earth? (Recall the Earth’s radius is about 3960 miles.)
  22. A small airplane is cruising directly westward at a low altitude northwest of Grand Junction. The plane’s altimeter is broken, so the pilot decides to do some manual calculations to determine the plane’s altitude. Looking out the cockpit windows, she spots the familiar roadways below to use as reference. Specifically she identifies 24 Rd and 23 Rd, which she knows are one mile apart. Using her trusty pocket inclinometer to measure the angle of depression below the plane’s horizontal flightpath, she determines that 24 Rd is located at 16.7° and 23 Rd is located at 11.81°. What’s the straight-line distance between the plane and 23 Rd? What’s the plane’s altitude?
  23. A park ranger wants to measure the width of a river along a particular segment. He finds a spot where the riverbanks are relatively straight, and sees a nice birch tree across the river. Pulling out his handy pocket-sized theodolite he determines that, from his location on the riverbank, his line of sight towards the birch makes a 81° angle with the riverbank. Walking 45′ along the river, past the spot directly across the river from the tree, to a large lingonberry shrub, he stops to measure again, determining that his line of sight towards the birch makes a 72° angle with the riverbank. What’s the distance between the lingonberry shrub and the birch tree? How wide is the river?
  24. A lineworker named Lionel needs to stabilize an old power pole. He plans to attach to guy-wires in line with the pole, securing the tip of the pole to the ground. Lionel decides to use some pre-cut wires that have been sitting in the back of his truck for weeks. One wire is 57′ and the other is 63′ After securing the guy-wires, Lionel measures that the angle the 63′ wire makes with the ground is 38°. How are apart are spots where the wires are anchored in the ground?
  25. Elijah is hiking up mountain at a constant incline of 16°, directly towards the rising sun, which at the moment is elevated at an angle of 64° above the horizon — that is above the horizontal, not the incline of the mountain. Suddenly Elijah finds himself in the cool shade of something tall in front of him, but his relief is cut short when he notices that it’s a bear. A polar bear has stood up on its hind legs to scope out Elijah as a potential snack, the tip of its head just blocking the sun from Elijah’s eyes. Elijah knows that the average polar bear is about 12′ tall when standing on its hind legs, and that he’s 5′6″ tall. How far away is the polar bear standing from Elijah?
  26. The leaning tower of Pisa leans at about a 4° angle — it used to be much worse, but folks have been working on stabilizing the tower. Anyways, a US tourist is visiting the tower and does a very natural thing for a character in a math problem to do: starting at the base of the tower, he walks exactly 200′ away from the tower in the direction that it’s leaning, then, turning around and pulling out his inclinometer, notes that the angle of his gaze to the highest point he sees on the tower is 45°. How tall would would the tower be if it were standing upright? Note the height of the tourist is negligible compared to the height of the tower.
  27. Two cars on a salt flat in Utah are parked with their back bumpers touching. In an instant they each take off in straight trajectories that are 130° apart from each other, one car travelling at 100 mph and the other at 110mph. How far apart are the cars after one hour?
  28. Rafael wants to install a slack line over a canyon. He scouts out some good spots on either side of the canyon to tether the line, but now needs to figure out how far apart they are. Using a total station from a fixed vantage point, he measures that his distance to each spot is 345′ and 551′ and that the angle between those spots from his vantage points is 43°. How far apart are the two tether spots?
  29. A plane is flying at a constant speed of 550 mph. After flying for one hour the plane makes a course correction, changing its heading by 7° before flying for two hours in this new direction. How far is the plane from its starting location?
  30. A plane is flying over the ocean at a constant altitude along a straight path, when the pilot spots two ships on the ocean below lying on the intended track of the plane (i.e. the plane will fly directly over each of them). The radar aboard the plane indicates that one ship is 8 miles away at a angle of depression of 78°, and the other is 13 miles away at a angle of depression of 50°. How far apart are the two ships? What is the altitude of the plane?
  31. A crude drawing of the San Francisco Bay with two vectors pointing in different directions westward. Two ships sail from west from the San Francisco bay at the same time, one at a speed of 24 knots (nautical miles per hour) with a heading of N 67° W, and the other at a speed of 29 knots with a heading of S 41° W. Ignoring the curvature of the earth, how far are the ships from each other after six hours? How far apart are they considering the earth’s curvature?
  32. Portobello the pirate is sailing from island to island within a small archipelago searching for treasure. One morning, starting from his camp on the beach of the prettiest of the islands, A crude drawing to three islands arranged roughly as described and two direction vectors indicating the path of travel he sails 1.1 miles at a heading of N 14° E towards a small sandy island to search. Having found nothing on that island by midday he then sails 0.8 miles at a heading of N 77° E towards a small overgrown island to search. Having found nothing there either by dusk he decides to return to his camp. What direction and how far should he sail to get back?
  33. A small triangular plot of land bound by three roads has sides that measure 44′ and 65′ and 53′. What are the measures of the angles at the corners of plot of land? What is the area of that plot of land?
  34. The GPS coordinates of a location on the surface of the earth are based on the measure of the angle that location makes with the earth’s center. A location’s latitude is its angular distance north-south from the earth’s equator, and a location’s longitude is its angular distance east-west from the prime meridian, which is the meridian passing through the Royal Observatory in Greenwich, London. For example, the city of Grand Junction CO has a latitude of 39°05′16″N and a longitude of 108°34′05″W. Recall that the radius of the earth is approximately 3960 miles.
    1. How far along the surface of the earth would a person have to travel south from Grand Junction before they reached the equator?
    2. The city of Cincinnati, Ohio has approximately the same latitude as Grand Junction but a longitude of 84°30′45″W. How far apart along the surface of the earth are Grand Junction and Cincinnati?
    3. The city of Pyongyang, North Korea has approximately the same latitude as Grand Junction but a longitude of 125°44′51″E. How far apart along the surface of the earth are Grand Junction and Pyongyang?
  35. Given three numbers \(\alpha\) and \(\beta\) and \(\gamma\) that are measures of the internal angles of a triangle, that triangle is uniquely determined up to similarity (AAA). But not all triples of numbers are the internal angles of a triangle; how can you tell which triples of numbers can and can’t be the measures of a triangle’s angles?
  36. Up to congruence, there is a unique triangle with side-lengths \(7\) and \(8\) and \(9.\)
    1. Calculate the measure of the angles internal to this triangle, calculate the area of this triangle, and make an accurate drawing of this triangle.
    2. In general given three numbers \(A\) and \(B\) and \(C\) that are the side-lengths of a triangle, that triangle is uniquely determined up to congruence (SSS). But not all triples of numbers are the side-lengths of a triangle; how can you tell which triples of numbers can and can’t be the side-lengths of a triangle?

  37. Up to congruence, there is a unique triangle with side-lengths \(7\) and \(8\) and an angle between those sides that measures \(56°.\) Calculate the length of the remaining side and measures of the remaining angles internal to this triangle, calculate the area of this triangle, and make an accurate drawing of this triangle.

    Note that any triple of numbers \(A\) and \(\gamma\) and \(B\) uniquely define a triangle with side-lengths \(A\) and \(B\) with an angle of measure \(\gamma\) between them (SAS), allowing for “degenerate” triangles when \(\gamma\) is a multiple of 180°.

  38. Up to congruence, there is a unique triangle with side-length \(7\) and angles adjacent to that side measuring \(34°\) and \(56°.\)
    1. Calculate the lengths of the remaining sides and measure of the remaining angle internal to this triangle, calculate the area of this triangle, and make an accurate drawing of this triangle.
    2. In general given three numbers, a side-length of a triangle \(B\) and the measures of two angles adjacent to that side \(\alpha\) and \(\gamma,\) that triangle is uniquely determined up to congruence (ASA). But not all such triples of numbers determine a triangle uniquely; how can you tell which side-lengths \(B\) and angles \(\alpha\) and \(\gamma\) really do or don’t uniquely determine a triangle?
  39. Up to congruence, there are two distinct triangles with side-lengths \(7\) and \(8\) and an angle adjacent only to the latter side measuring \(56°.\)
    1. Determine the these two possible triangle, and for each possibility calculate the length of the remaining side and measures of the remaining angles internal to the triangle, calculate the area of the triangle, and make an accurate drawing of the triangle.
    2. In general given three numbers, two side-lengths of a triangle \(A\) and \(B\) and the measure of an angle adjacent to the latte side \(\alpha,\) that triangle is sometimes uniquely determined up to congruence (SSA), but sometimes there is a second distinct triangle with those same measurements. Furthermore, not all such triples of numbers determine a triangle at all. How can you tell which side-lengths \(A\) and \(B\) and angles \(\alpha\) determine a unique triangle, determine two distinct triangles, or determine no triangle at all?
  40. exactly as the prompt describes Consider three circles having radii measuring four, five, and six, that are each mutually tangent to each other. What is the area of the small region bound by these three circles?
  41. What is the distance between the points in the plane with polar coordinates \((3,45°)?\) and \((5,210°)?\) In general, given two points with polar coordinates \((r_1, \theta_1)\) and \((r_2, \theta_2),\) what is a formula for the distance between them?
  42. Suppose the cos and tan buttons on your calculator are broken — only sin works — but you need to know a decimal approximation for the value of \[\csc(42°)+\cos^2(42°)-\tan(42°).\] How do you calculate this number using only the sine function?
  43. A right-angled halway corner as described in the prompt Suppose you have to transport a bunch of long, heavy, cast iron pipes one-by-one down a 7′ wide hallway, around a right-angled corner, and into a 5′ wide hallway. The pipes are awkward to carry so you decide to balance them on your shoulder, perfectly level with the ground, lest the pipe falls and damages the flooring. You realize that getting longer pipes around the corner is going to be tough, and will be downright impossible if the pipe is too long. What’s the longest length of a pipe that you could possibly swivel around that corner balanced on your shoulder?

  44. Standing at a point on the earth, how far east of you is a point that is one mile northeast of you? How far east of you is a point that is one mile east-northeast of you? How far east of you is a point that is one mile east-by-north of you? How far east of you is a point that is one mile in the direction halfway between east-by-north and east of you?

    More abstractly, what is a formula, in terms of \(n,\) for \(\cos\bigl(\frac{\pi}{2^n}\bigr)?\)

  45. What is the exact value, expressed in terms of radicals, of \(\cos\bigl(\frac{\pi}{5}\bigr)\) and \(\sin\bigl(\frac{\pi}{5}\bigr)?\)